preferred numbers

Objects are often manufactured in a series of sizes of increasing magnitude. The manufacturer must decide what those sizes should be. An example of such a series is one often used for money and packaging: 1, 2, 5, 10. The appeal of the series is its relation to decimalized currency; all the numbers divide evenly into ten. There is no three or thirty dollar bill.

In selecting the numbers for a series, people like numbers easily expressed in words; whole numbers for example. Formerly there was also a strong preference that the highest number in a series have a lot of prime factors, as 12 or 60 do, but the spread of numeracy and especially electronics have diluted this preference. Decimals have won out. Before the 20ᵗʰ century, many series of sizes proceeded by doubling, but the original, 18ᵗʰ century rules for the use of the metric system did not permit common fractions to be used with its units, though everyone does it (half a kilo, for example, is universal).

People also expect the difference between adjacent sizes to be constant. For this reason, if a series of sizes must cover a vast range, people are likely to want adjacent sizes to differ by a constant factor, making a geometrical or exponential series, such as 3, 6, 12. The alternative is a series in which adjacent numbers differ by a constant amount (an arithmetic series, such as 3, 5, 7). Shoe sizes, for example, are an arithmetic series in all cultures simply because feet vary only over a small range.

A drawing of circles within an angle. The circles have relative diameters of 10, 16, 25, 40, 63 and 100. The circles do not overlap. They just touch each other, and just touch the lines forming the angle.

If the first size is 10, a geometrical series can reach 100 in 5 steps by making adjacent sizes differ by a factor of , that is, multiply the preceding size by about 1.58.

A drawing of six circles with their centers on a line, drawn so that they just touch. The circles have relative diameters 10, 28, 46, 64, 82 and 100.

If the first size is 10, an arithmetical series can reach 100 in 5 steps by making adjacent sizes differ by 18, that is, add 18 to the preceding size.

A drawing of circles within an angle, drawn so that they just touch the lines forming the angle. The circles have relative diameters 10, 28, 46, 64, 82 and 100. The circles overlap.

An arithmetical series of sizes results in small sizes that are too far apart, and big sizes that are too close together.

Internationally Standardized Series

etching of military balloon operation

In 1877 a French military engineer, Col. Charles Renard (1849–1905) was given the job of improving captive balloons. (In those days armies used such balloons to observe the enemies' position.) He discovered that 425 different sizes of cable were being used to moor the balloons, a logistical nightmare, and set about determining how best to reduce these to a smaller number of sizes.

After determining that the relevant characteristic of the cable was its mass per unit length, Renard succeeded in replacing the 425 sizes with 17 sizes that covered the same range. To do this he made the sizes a geometric series in which by every fifth step the mass per unit length of the cable increased by a factor of ten:

a series: a, a times the fifth root of ten, a times the fifth root of ten squared, a times the fifth root of ten cubed, a times the fifth root of ten to the fourth power, and 10 a

which gives

a, 1.5848a, 2.5119a, 3.9811a, 6.3096a, 10a

If we let a = 10 and round off to whole numbers, we get the series

10, 16, 25, 40, 63, 100

which cinematographers will recognize as the old sequence of lens focal lengths for 16mm motion picture cameras.

The ratio between adjacent terms in a geometric series does not have to be based on a root of 10. For example, in the western musical scale, every twelfth term in the series increases the frequency by a factor of two, so the ratio between any two adjacent terms (i.e., notes) is

In the series “tablespoon, fluid ounce, quarter-cup, gill, cup, pint, quart, pottle, gallon,” the ratio between adjacent terms is 2, which was by far the most common ratio before the rise of decimal calculation.

To users of a decimally-oriented system of units, such as SI, Renard's series is much more useful than these other geometric series, because it begins on 10 and ends on 100. The ISO adopted Renard's series as the basis of the preferred numbers for use in setting metric sizes. The designations of the series they have defined begin with “R” as a tribute to Renard, and the series are called “renard series.”

The ISO has defined four basic series of preferred numbers:

R5: 10, 16, 25, 40, 63, 100.
R10: 10, 12.5, 16, 20, 25, 31.5, 40, 50, 63, 80, 100.
R20: 10, 11.2 12.5, 14, 16, 18, 20, 22.4, 25, 28, 31.5, 35.5, 40, 45, 50, 56, 63, 71, 80, 90, 100.
R40: 10, 10.6, 11.2, 11.8, 12.5, 13.2, 14, 15, 16, 17, 18, 19, 20, 21.2, 22.4, 23.6, 25, 26.5, 28, 30, 31.5, 33.5, 35.5, 37.5, 40, 42.5, 45, 47.5, 50, 53, 56, 60, 63, 67, 71, 75, 80, 85, 90, 95, 100.

It also defined an exceptional R80 series, with 81 values, which is little used.

Rounded series are also defined; they are indicated by a prime mark following the R.

R′10: 10, 12.5, 16, 20, 25, 32, 40, 50, 63, 80, 100.
R′20: 10, 11, 12.5, 14, 16, 18, 20, 22, 25, 28, 32,36, 40, 45, 50, 56, 63, 71, 80, 90, 100.
R′40:10, 10.5, 11, 12, 12.5, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 30, 32, 34, 36, 38, 40, 42, 45, 48, 50, 53, 56, 60, 63, 67, 71, 75, 80, 85, 90, 95, 100.

An even more rounded series is indicated by putting two prime marks after the R:

R″5: 10, 15, 25, 40, 60, 100.
R″10:10, 12, 15, 20, 25, 30, 40, 50, 60, 80, 100.
R″20:10, 11, 12, 14, 16, 18, 20, 22, 25, 28, 30, 35, 40, 45, 50, 50, 55, 60, 70, 80, 90, 100.

These preferred number series may be converted to sizes in several ways. Often they are multiplied or divided by some multiple of ten (e.g., 1000, 1600, 2500, 4000, 6300, 10000; 1.0, 1.6, 2.5, 4.0, 6.3, 10). Sometimes a portion of the range is chosen, in which case in the series is designated by including the two ends of the range in parentheses after the R number, like this: “R40 (14..20).”

Sometimes a series of sizes may be chosen by taking every other value in a series, or every third, or fourth, or so on. In the designation of such a selection the size of the skip is shown by a number after a slash. “R5/3,” for example, means a series consisting of every third value in the R5 series. Such a designation must include at least one end of the series. For example, R5/2 (10...400) would mean the series 10, 25, 63, 160, 400.

standards

ISO 3-1973, Preferred Numbers – Series of Preferred Numbers.

ISO 17-1973, Guide to the Use of Preferred Numbers and of Series of Preferred Numbers.

ISO 497-1973, Guide to the Choice of Series of Preferred Numbers and of Series Containing More Rounded Values of Preferred Numbers.

ANSI Z17.1-1973, American National Standard for Preferred Numbers.

for further reading

A. Van Dyck.
Preferred numbers.
Proceedings of the Institute of Radio Engineers, volume 24, pages 159-179 (February 1936)

C. F. Hirshfeld and C. H. Berry.
Size standardization by preferred numbers.
Mechanical Engineering, vol. 44, no. 12 (Dec. 1922), page 791.